Problem 11 - Technical Mathematics with Calculus

Question

Dividing Approximate Numbers Divide, and then round your answer to the proper number of digits. $$94,840 \div 1.33876$$

Step-by-Step Solution

Answer

Divide 94,840 by 1.33876 to get approximately 70839.24219, then round it to five significant figures, resulting in 70,839.

Step 1: Perform the Division

Using a calculator or long division, divide 94,840 by 1.33876. This will give you a lengthy decimal value as the result.

Step 2: Determine the Number of Significant Figures

Given that the divisor is an approximate number with five decimal places (1.33876), you should round the result of the division to the least number of significant figures in either of the original numbers.

Step 3: Round to the Proper Number of Significant Figures

The original numbers, 94,840 and 1.33876, have five significant figures each. Since both have the same number of significant figures, round the result of the division to five significant figures.

Key Concepts

Significant Figures

When working with measurements and calculations, accuracy and precision are of great importance. This is where the concept of significant figures comes into play. Significant figures, or 'sig figs', are the digits in a number that carry meaningful contribution to its precision. They include all non-zero digits, any zeros between significant digits, and any trailing zeros in the decimal portion.

For example, in the number 1.33876, all digits are significant because they represent precision in measurement. This importance becomes apparent during division; the quotient must be trimmed to maintain the level of precision from the least precise number involved in the operation. Typically, if you're dividing two numbers, the result should be presented with the same number of significant figures as the number with the fewest significant figures in the initial problem. In other words, significant figures keep our results consistent with the original data's precision levels.

Decimal Division

Decimal division can sometimes seem intimidating, but it follows the same basic principles as dividing whole numbers. When dividing numbers such as 94,840 by 1.33876, you're essentially finding how many times the divisor fits into the dividend. You can use long division or a calculator to find this value, which initially yields a long decimal number.

During manual calculations, aligning the decimal points is key. When the divisor is a decimal, you can make it a whole number by shifting the decimal point to the right; however, you must do the same to the dividend to keep the balance. This action does not change the quotient but simplifies the division. The number of digits after the decimal point in the quotient will often exceed the level of significance from our original numbers—this is when rounding comes into play.

Rounding Decimals

Rounding decimals is a necessary step in many mathematical operations, particularly when the number of decimal places is excessive or when we need to adjust the precision of an answer to match the significant figures in a problem. After division, we are often left with a long string of digits after the decimal point, many of which may not be significant. Depending on the context, we round them to a certain number of decimal places.

To round a decimal, we look at the digit following the last significant figure we want to keep. If that digit is five or higher, we increase the final significant figure by one; if it is less than five, we leave it as is. In the example of dividing 94,840 by 1.33876, once we have our lengthy decimal answer, we round it to five significant figures, preserving the precision indicated by the original measurements. This not only communicates the accuracy of our calculation but also prevents the introduction of error from insignificant digits.